- #1
ChemEng1
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Homework Statement
Let [itex]\left\{E_{k}\right\}_{k\in N}[/itex] be a sequence of measurable subsets of [0,1] satisfying [itex]m\left(E_{k}\right)=1[/itex]. Then [itex]m\left(\bigcap^{\infty}_{k=1}E_{k}\right)=1[/itex].
Homework Equations
m denotes the Lebesgue measure.
"Measurable" is short for Lebesgue-measurable.
The Attempt at a Solution
My intuition was to construct 2 disjoint and uncountable subsets of an uncountable set that all have measure equal to 1.
After much struggling, I don't think this is possible. To be able to construct such a counterexample would mean there is a mapping of each subset to the natural numbers which would make them both countable.
Are there non-constructive approaches to building this counterexample? My attempts haven't yielded anything meaningful. Each example (rational and irrational, transcendental and algebraic) I've considered ends up with measure 1 and measure 0 subsets instead of both measure 1.
I am not sure if my counterexample building skills are lacking or I just need to adjust my intuition. Any help would be appreciated.